a) We assume k>0, and look for a solution u(t,x)=U(w(eps)t,x), where U=U0+eps*U1+eps^2*U2+O(eps^2), and w(eps)=1+eps*w1+eps^2*w2+O(eps^3).
We assume u(0,x)=A*cos(x)+eps*B*cos(3x)+O(eps^2), and u_t(0,x)=0. A~=0, and U_j is bounded for all j.

We look for a positive value k0 of k, for which not all resonant terms can be eliminated from the equation for U2. Show that for k~=k0 all resonant terms have been eliminated from the equations for U1 and U2.

b) Again assume that k>0. Let u(t,x,eps) be a solution of the PDE that is periodic in x with period 2*pi, periodic in t with minimal period P(eps) and is even in x and in t. Assume that as eps -> 0 the solution u(t,x,eps) converges to a nonzero function u0(t,x) and its period P(eps) converges to a nonzero P0. Show that there is a countable set E of positive real numbers such that if k is not in E then there exist a nonzero A and an integer n such that u0(t,x)=A*cos(sqrt(n^2+k^2)*t)*cos(n*x).

c) Is k0 from (a) is in E from (b)?

This would mean the world if anyone would help me!
You can also contact me by mail:w.jhon1984@gmail.com

What puzzles me is that the theorem works when the base is a prime in the ring
of Gaussian integers and the exponent is a prime of shape 4m + 1
but does not work when the exponent is a prime of shape 4m+3.Can
any one throw some light on this?